快速多极边界元方法在二维声散射问题中的应用

摘要

The fast multi-pole method (FMM) is a very effective approach to accelerating numerical solutions of the boundary element method (BEM) for the problems requiring large scale computation. An application of the FMM to two-dimensional boundary integral equation method for the acoustic scattering problem was discussed. We seek the solution of Helmholtz equation Δu+k2u=0 in the form of a combined single- and double-layer potential. The boundary integral equation is discretized with Nystrm method. It is obvious that the kernel of integral operator is unsymmetrical. If the resulting linear system is solved by the conjugate gradient method of unsymmetrical linear system, both the products of matrix A with vector x and AT with x should be repeatedly evaluated. The hierarchical cell structures of FMM with two different methods was constructed, and the multi-pole expansion, local expansion and translations of the coefficients were given for the second integral operator A and its conjugate operator AT. The boundary integral equation was solved by FMM. The numerical results show that FMM is more efficient than the direct computation approach.%快速多极算法(FMM)是求解边界元方法(BEM)在大尺度情况下的一种非常有效的算法.研究了快速多极算法在二维声散射问题的边界积分方程求解中的应用.给出了积分核函数以及其共轭积分算子核函数的多极展开式,局部展开式以及相应展开系数之间的转化关系.分别应用两种不同的层级树结构的FMM来进行求解,并对两种树结构下的求解效率进行了对比.数值算例表明用快速多极算法求解该问题时在存储量和计算量上比直接求解方法效率更高.